Question: Consider the line parameterized by
\begin{align*} 
x&= 4t + 2,\\ 
y& = t+2.
\end{align*}Find a vector $\begin{pmatrix}a \\ b \end{pmatrix}$ pointing from the origin to this line that is parallel to $\begin{pmatrix}2 \\1 \end{pmatrix}$.
Answer: Here is a plot of the line:

[asy]
size(200); 
import TrigMacros; 
//Gives the maximum line that fits in the box. 
path maxLine(pair A, pair B, real xmin, real xmax, real ymin, real ymax) 
{
    pair[] endpoints = intersectionpoints(A+10(B-A) -- A-10(B-A), (xmin, ymin)--(xmin, ymax)--(xmax, ymax)--(xmax, ymin)--cycle);
    if (endpoints.length >= 2) return endpoints[0]--endpoints[1];
    else return nullpath;  
}

rr_cartesian_axes(-3, 9, -3, 6,complexplane=false,usegrid=true);

pair A = (2, 2); 
pair B = (6,3); 
draw(maxLine(A, B, -3, 9, -3, 6)); 
[/asy]
We need a vector pointing from the origin to the line in the direction of $\begin{pmatrix}2\\1\end{pmatrix}$. That means that the tail of the vector will be at the origin, and the head of the vector will be somewhere on this blue line:

[asy]
size(200); 
import TrigMacros; 
//Gives the maximum line that fits in the box. 
path maxLine(pair A, pair B, real xmin, real xmax, real ymin, real ymax) 
{
    pair[] endpoints = intersectionpoints(A+10(B-A) -- A-10(B-A), (xmin, ymin)--(xmin, ymax)--(xmax, ymax)--(xmax, ymin)--cycle);
    if (endpoints.length >= 2) return endpoints[0]--endpoints[1];
    else return nullpath;  
}

rr_cartesian_axes(-3,9,-3,6,complexplane=false,usegrid=true);

pair A = (2, 2); 
pair B = (6,3); 
draw(maxLine(A, B, -3, 9, -3, 6)); 
draw(maxLine((0,0), B, -3, 9, -3, 6), blue); 
[/asy]
Since the head of the vector needs to be on the black line as well, it must be the intersection point of the two lines.

The lines intersect when
\[\begin{pmatrix} x \\ y \end{pmatrix} = k \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 2k \\ k \end{pmatrix}\]for some real number $k.$  In other words, $4t + 2 = 2k$ and $t + 2 = k.$  Solving, we find $t = 1$ and $k = 3.$  Therefore, the lines intersect at $\boxed{\begin{pmatrix}6\\3\end{pmatrix}}.$

[asy]
size(200); 
import TrigMacros; 
//Gives the maximum line that fits in the box. 
path maxLine(pair A, pair B, real xmin, real xmax, real ymin, real ymax) 
{
    pair[] endpoints = intersectionpoints(A+10(B-A) -- A-10(B-A), (xmin, ymin)--(xmin, ymax)--(xmax, ymax)--(xmax, ymin)--cycle);
    if (endpoints.length >= 2) return endpoints[0]--endpoints[1];
    else return nullpath;  
}

rr_cartesian_axes(-3,9,-3,6,complexplane=false,usegrid=true);

pair A = (2, 2); 
pair B = (6,3); 
draw(maxLine(A, B, -3, 9, -3, 6)); 
draw((0,0)--B, red, Arrow(size = 0.3cm)); 
[/asy]